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Rough Isometries of Order Lattices and Groups Rough Isometries of Order Lattices and Groups Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universit¨at G¨ottingen vorgelegt von Andreas Lochmann aus Bad Pyrmont G¨ottingen 2009 Referenten der Dissertation: Prof. Dr. Thomas Schick Prof. Laurent Bartholdi, Ph.D. Mitglieder der Pr¨ufungskommission: Prof. Dr. Thomas Schick Prof. Laurent Bartholdi, Ph.D. Prof. Dr. Rainer Kress Prof. Dr. Ralf Meyer Prof. Dr. Ulrich Stuhler Prof. Dr. Andreas Thom Tag der m¨undlichen Pr¨ufung: 6. August 2009 Gewidmet meinen Eltern Preface Assume you measure the distances between four points and find the following: One of the distances is 143, the distance between the other pair of points is 141, and all other distances are 99. You know that your error in measurement is not greater than 1. Is this result then compatible with the hypothesis, that the four points constitute a square in Euclidean space? Another example: You distribute five posts equidistantly along straight rails, then watch a train driving on the rails and take the times at which it passes the posts. The times you measure are: 0, 100, 202, 301, 401. However, the time the clocks show might deviate by up to 1 from the clock at the starting post. Are the measured times consistent with the assumption that the train has constant speed in Newtonian mechanics? These two puzzles are metaphors for a general problem in the application of mathematics. We have to use measurements that might contain errors to verify predictions of an ideal theory. If, for example, you find a function f between vector spaces to fulfill linearity only approximately, i.e. there is ǫ > 0 such that d f(x + y), f(x) + f(y) ǫ ≤ for all x, y; can you find a perfectly linear function g which is near f? Ulam stated this question during a talk around 1941, and Hyers found a positive solution shortly after by applying scaling arguments ([Hy]). Ulam and Hyers continued their collaboration, and were able to give positive answer in [HU] to the following question in 1947: Is every ǫ-isometric homomorphism of real- valued continuous function spaces in bounded distance to an isometry? Theorem 1 (Hyers-Ulam) 1 Let K, K′ be compact metric spaces and let E and E′ be the spaces of all real valued continuous functions on K and K′, respectively. Let T (f) be a homo- morphism of E onto E′ which is also an ǫ-isometry for some arbitrary, fixed ǫ 0. Then there exists an isometric transformation U(f) of E onto E such ≥ ′ that U(f) T (f) 21ǫ for all f in E. In particular, the underlying metric || − || ≤ spaces K and K′ are homeomorphic. This theorem evoked several similar results for linear functions ([Ra2]), but also about stability of differential equations, the stability of group actions ([GKM]), and approximate group homomorphisms, as defined by Ulam in [U], section VI.1; see [HR] for a survey on this topic. 6 It is interesting to see that the isometry of the function spaces implies con- ditions of their underlying metric spaces—in this case they are homeomorphic. Our main interest during a large part of this thesis will be function spaces of Lipschitz functions, and their stability against rough isometries. Rough isometries are in some sense the prototype of approximative struc- ture: the description of Hyers’ ǫ-linear functions would not be possible without a metric on the vector spaces. The idea simply is to replace an identity x = y by d(x, y) ǫ. In this sense, each description of approximations needs metric ≤ spaces. As the natural mappings of metric spaces are Lipschitz maps, isome- tries, and isometric embeddings, their rough counterparts are the most natural approximative tools, among them the rough isometry (or ǫ-isometry) as mea- sure to describe the similarity of metric spaces. Our goal is to shed some light on the interplay of rough isometries with order lattice and group structures, but in particular to demonstrate their use in the study of the geometry of Lipschitz function spaces. There are detailed books and various articles on many aspects of Lipschitz function spaces, including isometries between them. A nice survey is Weaver’s book on Lipschitz Algebras [Wv] (cf. Section 2.6, where Weaver elaborates on the exact same questions we want to tackle here, but in a non-coarse context and under different conditions). However, to the knowledge of the author, no book nor article dealt with their coarse geometry yet. On the other hand, Lipschitz functions naturally appear in many aspects of coarse geometry, such as the Levy concentration phenomenon or the definition of Lipschitz-Hausdorff distance in [Gv2]. But they are not dealt with as metric spaces either. The structure of this thesis is as follows: We first give an overview of basic notions in handling with metric spaces and order lattices. We assume that the reader is familiar with metric spaces, so this part will just cover our use of infinities in metric spaces, the definition of hyperconvex and injective metric spaces, and those notions from coarse geometry which apply to our situation. The first chapter deals with a generalization of Birkhoff’s metric lattices to put the supremum metric on Lipschitz function spaces into a common context. During this chapter we derive several useful tools for to handle distances in Lipschitz function spaces, and introduce several metrics for Lipschitz functions, familiar ones as well as some more exotic distance functions. The final section in this chapter will define a very important notion for our analysis, a version of irreducibility which depends on the chosen metric of a lattice. We further see some first, general properties of this new kind of irreducibility. The second chapter will concentrate on Lipschitz function spaces with supre- mum metric, and we will examine the question, whether they are roughly iso- metric when their underlying spaces are. We first take a look at a simple, but efficient smoothening algorithm, then take a digression to demonstrate a corresponding Lemma for hyperconvex-space-valued Lipschitz functions. We then revive the situation of the first chapter, and define a rough ver- sion of isomorphism for intervaluation metric lattices, which fits well into the context of our new irreducibility-condition. By introducing minimal Lipschitz functions (which we call “Λ-functions”), we provide a lattice-theoretic base of the Lipschitz function space (see Definition 76), and put the smoothening re- Preface 7 sult of the first section into this more advanced context, re-proving it using new techniques: Theorem 2 2 Let X,Y be metric spaces, ǫ 0. For each ǫ-isometry η : X Y , there is a + ≥ → 4ǫ-ml-isomorphism κ : Lip Y Lip X such that κ is ǫ-near f f η for all → 7→ ◦ f Lip Y (see Definitions 6, 29, 80, 89). ∈ We then continue to analyze our Λ-functions and find them to be exactly the irreducible elements of the first chapter. This allows us to state a converse of Theorem 2: Theorem 3 3 Let X,Y be complete metric spaces and ǫ 0. For each ǫ-ml-isomorphism + ≥ κ : Lip Y Lip X there is an 88ǫ-isometry η : X Y , such that κ is 62ǫ-near → → f f η for all f Lip Y . 7→ ◦ ∈ As we make use of the order-lattice structure of the Lipschitz function spaces in our proofs, the following theorem by Kaplansky [Kp] is related to our results as well and we state it in its formulation by Birkhoff ([Bi1], 2nd ed. p. 175f.): Theorem 4 (Kaplansky) 4 Any compact Hausdorff space is (up to homeomorphism) determined by the lattice of its continuous functions. However, the proofs we present here not only make use of the lattice struc- ture of Lip X, but of a metric on it as well. In this sense, the comparison with Kaplansky’s Theorem 4 is not perfect. We then ask whether it is possible to state Theorems 2 and 3 in a more general context, and give counter-examples to a broad range of different metrics on Lipschitz function spaces, among them the seemingly convenient L1-metric. We close this chapter with an example concerning quasi-isometries, and an application in the theory of scaling limits. The third chapter discusses the use of rough isometries in the context of finitely generated groups. One cannot overvalue the importance of coarse meth- ods in Geometric Group Theory, which is in large parts based on the notion of quasi-isometry. We begin this chapter with another digression, demonstrating how some proofs can be reformulated in a rough context, while this is not possi- ble with others. We then try to point out the possibilities rough isometries add to Geometric Group Theory, by providing a specifically rough isometry invariant (the exponential growth rate), and then show the connection between symme- tries in some virtually abelian groups and rough isometries between them. We then continue with a theorem motivated by Bartholdi and translate it into the rough context in the final section, where we show that uniformly rough and quasi-isometries imply commensurability. Finally a remark to our notation: N0 denotes the non-negative integers, N∗ the positive integers, Cn denotes the cyclic group of order n. We sometimes drop function brackets where feasible. 8 Acknowledgements. The author wants to thank the “Graduiertenkolleg Gruppen und Geometrie” for supporting his research financially, through con- ferences and workshops, and by supporting the great open atmosphere at the institute. Best thanks go to Prof. Dr. Thomas Schick, Prof. Laurent Bartholdi, and Prof.
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